1. An Algorithm for the Traveling Salesman Problem John D. C. Little, Katta G. Murty, Dura W. Sweeney, and Caroline Karel 1963 Speaker: Huang Cheng-Kang

2. What ’ s TSP? A salesman, starting in one city, wishes to visit each of n – 1 other cities once and only once and return to the start. In what order should he visit the cities to minimize the total distance traveled?

3. The Algorithm The basic method: Branch – break up the set of all tours Bound – calculate a lower bound

4. Notation (1/2) The entry in row i and column j of the matrix is the cost for going from city i to city j. Let A tour, t, can be represented as a set of n ordered city pairs, e.g., cost matrix

5. Notation (2/2) The cost of a tour, t, under a matrix, C, is the sum of the matrix elements picked out by t and will be denoted by : Also, let nodes of the tree; a lower bound on the cost of the tours of X, i.e., for t a tour of X; the cost of the best tour found so far. in t

6. Lower Bounds The useful concept in constructing lower bounds will be that of reduction. t = [(1,2) (2,3) (3,4) (4,1)] z = 3+6+6+9 = 24 Reduce Reduce Concept: at least one zero in each row and column z = 16+(3+8) = 24

7. Branching all tours

8. 4 1 0 1 6 6 the sum of the smallest element in row i and column j :

9. 下限 = 17

10. all tours 16 1722

11. 4 01 0

12. all tours 16 1722 17 21

13. all tours 16 1722 17 21 17 無解 17 t = [(1,2) (2,3) (3,4) (4,1)] z = 3+6+6+9 = 24